Theorem imbi1d

Description: Deduction adding a consequent to both sides of a logical equivalence. (Contributed by NM, 11-May-1993) (Proof shortened by Wolf Lammen, 17-Sep-2013)

	Ref	Expression
Hypothesis	imbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
Assertion	imbi1d	$$⊢\phi \to \left(\left(\psi \to \theta \right)\leftrightarrow \left(\chi \to \theta \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		imbid.1	$$⊢\phi \to \left(\psi \leftrightarrow \chi \right)$$
2	1	biimprd	$$⊢\phi \to \left(\chi \to \psi \right)$$
3	2	imim1d	$$⊢\phi \to \left(\left(\psi \to \theta \right)\to \left(\chi \to \theta \right)\right)$$
4	1	biimpd	$$⊢\phi \to \left(\psi \to \chi \right)$$
5	4	imim1d	$$⊢\phi \to \left(\left(\chi \to \theta \right)\to \left(\psi \to \theta \right)\right)$$
6	3 5	impbid	$$⊢\phi \to \left(\left(\psi \to \theta \right)\leftrightarrow \left(\chi \to \theta \right)\right)$$